NAG FL Interface
e04lyf (bounds_​mod_​deriv2_​easy)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

On entry: the number $n$ of independent variables.

Constraint: $\mathbf{n}\ge 1$.

2: $\mathbf{ibound}$ – Integer Input

On entry: indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:

$\mathbf{ibound}=0$

If you are supplying all the $l_j$ and $u_j$ individually.

$\mathbf{ibound}=1$

If there are no bounds on any $x_j$.

$\mathbf{ibound}=2$

If all the bounds are of the form $0\le x_j$.

$\mathbf{ibound}=3$

If $l_1=l_2=\cdots =l_n$ and $u_1=u_2=\cdots =u_n$.

Constraint: $0\le \mathbf{ibound}\le 3$.

3: $\mathbf{funct2}$ – Subroutine, supplied by the user. External Procedure

You must supply this routine to calculate the values of the function $F\left(x\right)$ and its first derivatives $\frac{\partial F}{\partial x_j}$ at any point $x$. It should be tested separately before being used in conjunction with e04lyf (see the E04 Chapter Introduction).

The specification of funct2 is:

Fortran Interface

Subroutine funct2 ( n, xc, fc, gc, iuser, ruser) Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: xc(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fc, gc(n)

C Header Interface

void  funct2_ (const Integer *n, const double xc[], double *fc, double gc[], Integer iuser[], double ruser[])

C++ Header Interface

#include <nag.h> extern "C" { void  funct2_ (const Integer &n, const double xc[], double &fc, double gc[], Integer iuser[], double ruser[]) }

1: $\mathbf{n}$ – Integer Input

On entry: the number $n$ of variables.

2: $\mathbf{xc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input

On entry: the point $x$ at which the function and its derivatives are required.

3: $\mathbf{fc}$ – Real (Kind=nag_wp) Output

On exit: the value of the function $F$ at the current point $x$.

4: $\mathbf{gc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{gc}\left(\mathit{j}\right)$ must be set to the value of the first derivative $\frac{\partial F}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{j}=1,2,\dots ,n$.

5: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

6: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

funct2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to funct2.

funct2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: funct2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.

4: $\mathbf{hess2}$ – Subroutine, supplied by the user. External Procedure

You must supply this routine to evaluate the elements $H_{ij}=\frac{{\partial }^{2}F}{\partial x_i\partial x_j}$ of the matrix of second derivatives of $F\left(x\right)$ at any point $x$. It should be tested separately before being used in conjunction with e04lyf (see the E04 Chapter Introduction).

The specification of hess2 is:

Fortran Interface

Subroutine hess2 ( n, xc, heslc, lh, hesdc, iuser, ruser) Integer, Intent (In) :: n, lh Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: xc(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: heslc(lh), hesdc(n)

C Header Interface

void  hess2_ (const Integer *n, const double xc[], double heslc[], const Integer *lh, double hesdc[], Integer iuser[], double ruser[])

C++ Header Interface

#include <nag.h> extern "C" { void  hess2_ (const Integer &n, const double xc[], double heslc[], const Integer &lh, double hesdc[], Integer iuser[], double ruser[]) }

1: $\mathbf{n}$ – Integer Input

On entry: the number $n$ of variables.

2: $\mathbf{xc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input

On entry: the point $x$ at which the derivatives are required.

3: $\mathbf{heslc}\left(\mathbf{lh}\right)$ – Real (Kind=nag_wp) array Output

On exit: hess2 must place the strict lower triangle of the second derivative matrix $H$ in heslc, stored by rows, i.e., set $\mathbf{heslc}\left(\left(\mathit{i}-1\right)\left(\mathit{i}-2\right)/2+\mathit{j}\right)=\frac{{\partial }^{2}F}{\partial x_{\mathit{i}}\partial x_{\mathit{j}}}$, for $\mathit{i}=2,3,\dots ,n$ and $\mathit{j}=1,2,\dots ,\mathit{i}-1$. (The upper triangle is not required because the matrix is symmetric.)

4: $\mathbf{lh}$ – Integer Input

On entry: the length of the array heslc.

5: $\mathbf{hesdc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

On exit: must contain the diagonal elements of the second derivative matrix, i.e., set $\mathbf{hesdc}\left(\mathit{j}\right)=\frac{{\partial }^{2}F}{\partial x_{\mathit{j}}^2}$, for $\mathit{j}=1,2,\dots ,n$.

6: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

7: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

hess2 is called with the arguments iuser and ruser as supplied to e04lyf. You should use the arrays iuser and ruser to supply information to hess2.

hess2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04lyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: hess2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04lyf. If your code inadvertently does return any NaNs or infinities, e04lyf is likely to produce unexpected results.

5: $\mathbf{bl}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input/Output

On entry: the lower bounds $l_j$.

If ibound is set to $0$, $\mathbf{bl}\left(\mathit{j}\right)$ must be set to $l_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If a lower bound is not specified for any $x_j$, the corresponding $\mathbf{bl}\left(j\right)$ should be set to $-{10}^6$.)

If ibound is set to $3$, you must set $\mathbf{bl}\left(1\right)$ to $l_1$; e04lyf will then set the remaining elements of bl equal to $\mathbf{bl}\left(1\right)$.

On exit: the lower bounds actually used by e04lyf.

6: $\mathbf{bu}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input/Output

On entry: the upper bounds $u_j$.

If ibound is set to $0$, $\mathbf{bu}\left(\mathit{j}\right)$ must be set to $u_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If an upper bound is not specified for any $x_j$ the corresponding $\mathbf{bu}\left(j\right)$ should be set to ${10}^6$.)

If ibound is set to $3$, you must set $\mathbf{bu}\left(1\right)$ to $u_1$; e04lyf will then set the remaining elements of bu equal to $\mathbf{bu}\left(1\right)$.

On exit: the upper bounds actually used by e04lyf.

7: $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input/Output

On entry: $\mathbf{x}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$. The routine checks the gradient and the Hessian matrix at the starting point, and is more likely to detect any error in your programming if the initial $\mathbf{x}\left(j\right)$ are nonzero and mutually distinct.

On exit: the lowest point found during the calculations. Thus, if $\mathbf{ifail}=\mathbf{0}$ on exit, $\mathbf{x}\left(j\right)$ is the $j$th component of the position of the minimum.

8: $\mathbf{f}$ – Real (Kind=nag_wp) Output

On exit: the value of $F\left(x\right)$ corresponding to the final point stored in x.

9: $\mathbf{g}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

On exit: the value of $\frac{\partial F}{\partial x_{\mathit{j}}}$ corresponding to the final point stored in x, for $\mathit{j}=1,2,\dots ,n$; the value of $\mathbf{g}\left(j\right)$ for variables not on a bound should normally be close to zero.

10: $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer array Workspace

11: $\mathbf{liw}$ – Integer Input

On entry: the dimension of the array iw as declared in the (sub)program from which e04lyf is called.

Constraint: $\mathbf{liw}\ge \mathbf{n}+2$.

12: $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) array Workspace

13: $\mathbf{lw}$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which e04lyf is called.

Constraint: $\mathbf{lw}\ge \max \left(\mathbf{n}\times \left(\mathbf{n}+7\right),10\right)$.

14: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

15: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by e04lyf, but are passed directly to funct2 and hess2 and may be used to pass information to these routines.

16: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{ibound}=〈\mathit{\text{value}}〉$.
Constraint: $0\le \mathbf{ibound}\le 3$.

On entry, $\mathbf{ibound}=0$ and $\mathbf{bl}\left(\mathit{j}\right)>\mathbf{bu}\left(\mathit{j}\right)$ for some $j$.

On entry, $\mathbf{ibound}=3$ and $\mathbf{bl}\left(1\right)>\mathbf{bu}\left(1\right)$.

On entry, $\mathbf{liw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liw}\ge 〈\mathit{\text{value}}〉$.

On entry, $\mathbf{lw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lw}\ge 〈\mathit{\text{value}}〉$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

$\mathbf{ifail}=2$

There have been $50\times \mathbf{n}$ function evaluations.

The algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that $F\left(x\right)$ has no minimum.

$\mathbf{ifail}=3$

The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.

$\mathbf{ifail}=5$

It is probable that a local minimum has been found, but it cannot be guaranteed.

$\mathbf{ifail}=6$

It is possible that a local minimum has been found, but it cannot be guaranteed.

$\mathbf{ifail}=7$

It is unlikely that a local minimum has been found.

$\mathbf{ifail}=8$

It is very unlikely that a local minimum has been found.

$\mathbf{ifail}=9$

The modulus of a variable has become very large. There may be a mistake in your supplied routines, your problem has no finite solution, or the problem needs rescaling.

$\mathbf{ifail}=10$

It is very likely that you have made an error forming the gradient.

$\mathbf{ifail}=11$

It is very likely that you have made an error forming the 2nd derivatives.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results